engleski

Nonuniform spectrum on the half line and perturbations

For a one-sided nonautonomous dynamics defined by a sequence of invertible matrices, we develop a spectral theory (in the sense of Sacker and Sell) for the notion of a nonuniform exponential dichotomy with an arbitrarily small nonuniform part. We emphasize that this notion is ubiquitous in the context of ergodic theory, unlike the notion of a uniform exponential dichotomy. In particular, we show that each Lyapunov exponent belongs to one interval of the spectrum. We also consider a class of sufficiently small nonlinear perturbations of a linear dynamics satisfying a nonuniform bounded growth condition and we show that each solution is either eventually zero or the Lyapunov exponents belong to one interval of the spectrum.

Nonuniform hyperbolicity ; spectrum ; perturbations

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